Atlas › Modular forms › Weight 4 › Level 87 › 87.4.a

Newspace 87.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(87))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
87.4.a.a	2	$\mathbb{Q}(\sqrt{17})$	$2q - 5q^{2} + 6q^{3} + 5q^{4} - 11q^{5} - 15q^{6} - 24q^{7} + O(q^{8})$	− +	$-1$
87.4.a.b	2	$\mathbb{Q}(\sqrt{41})$	$2q - q^{2} - 6q^{3} + 5q^{4} - q^{5} + 3q^{6} - 24q^{7} + O(q^{8})$	+ −	$-1$
87.4.a.c	5	degree 5	$5q + 3q^{2} - 15q^{3} + 29q^{4} - q^{5} - 9q^{6} + 4q^{7} + O(q^{8})$	+ +	$+1$
87.4.a.d	5	degree 5	$5q + 3q^{2} + 15q^{3} + 29q^{4} + 29q^{5} + 9q^{6} + 4q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
87.4.a.a	$x^{2} + 16x - 4$
87.4.a.b	$x^{2} + 2x - 40$
87.4.a.c	$x^{5} - 2x^{4} - 636x^{3} + 3421x^{2} + 102980x - 835200$
87.4.a.d	$x^{5} - 32x^{4} + 196x^{3} + 1807x^{2} - 10718x - 46936$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(87))$	14
Old forms	14
Cusp forms $S_{4}(\Gamma_0(87))$	28
Eisenstein series $E_{4}(\Gamma_0(87))$	4
Modular forms $M_{4}(\Gamma_0(87))$	32

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(87)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.106 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 3, 29, 87]: [1, 2, 9, 32] (certified at ell = 2147483549)
T = [(2, 1), (5, 1)]; h of degree 14 = 1 x 14 (phi(ord eps) x dim S_k^new); 3 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(87, 4)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 87:4 1000

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